Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Is the Fourier transform of the function correct?

I am asked to calculate the Fourier transform of the function $$e^{-\frac{x^2}{2a^2}}, a>0$$ $$$$ After calculations I have found the following: $$a \sqrt{2 \pi} e^{-\frac{a^2 k^2}{2}}$$ $$$$ Is ...
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47 views

Need help to understand calculating fourier transform of 1

this issue is related to a physics problem, but since it is mathematical I will post it here. When calculating the following Fourier transform $$ -i\int_\infty^\infty dt~ e^{i\omega ...
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2answers
42 views

discrete fourier transform proof (show equals n*I)

Let $w=e^{(-2\pi i/n)}$. Let $W$ be an $n \times n$ matrix defined by $$ W = \begin{pmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & w & w^2 & w^3 & \cdots & ...
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22 views

Find the function $f(x)$ when it satisfies the ode

The Fourier transform of the function $\frac{d}{dx}f(x)+xf(x)$ is $i[\frac{d}{dk}\widetilde{f}(k)+k\widetilde{f}(k)]$, so if a function $f(x)$ satisfies the ode $\frac{d}{dx}f(x)+xf(x)=0$, then the ...
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105 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
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82 views

Evaluate the following Dirac delta integrals:

a) $ \int^{+\infty}_{-\infty} \delta'(t-\pi)e^{-t^2} \; dt$ b) $ \int^{+\infty}_{-\infty} \delta(-3t)(\frac{e^{-t^2}}{\ln(t^2 + 3)}) \; dt $ c) $ \int^{+\infty}_{-\infty} \delta(4t)\sinh{t^2} ...
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Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
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65 views

Calculate the Fourier Transform of the function

I have to calculate the Fourier transform of the function $f(x)=sgn(x) e^{-a |x|}, a \geq 0$. After that I have to take the limit $ a \rightarrow 0$ to calculate the Fourier transform of the sign ...
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2answers
69 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
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1answer
163 views

Density of a set of functions in Schwartz space

I have a difficulty doing the following problem: Let $S(\mathbb{R}^n)$ be the Schwartz space. I need to determine whether the following set of functions $A$: $$A= \{f\in S(\mathbb{R}^n): ...
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47 views

How can I prove this property?

I need some help... I am asked to prove the following property of the Fourier transform, when $F[f(x)]=\widetilde{f}(x)$, where $F[f(x)]$ is the Fourier transform of $f(x)$: $$F[ \widetilde{f}(x) ...
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1answer
64 views

The derivative of a function is square integrable assuming Fourier transform dominated

I am struggling in solving the second part of this problem. Let $g$ be a continuous function in $L^1(\mathbb{R})$ whose Fourier transform is the function $F$. Suppose $|F(x)|\leq (1+x^2)^{-2}$. Prove ...
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0answers
49 views

Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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2k views

Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
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103 views

Characteristic Function as Fourier Multiplier

In lecture notes I have, it is mentioned that the characteristic function $\chi_I$ of an interval in $\mathbb{R}$ is an $L^p$ Fourier multiplier for $1 < p < \infty$. I thought this would be ...
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1answer
60 views

Integral equation, Fourier transform

Find all functions $ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve $\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$, $ t\in \mathbb{R}$ How do I solve this? I know that the left part is the ...
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22 views

fourier-transform: where is my mistake?

I am trying to do a fourier-transform of the function $$\psi(x,t=0) = \frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}e^{-\frac{x^2}{4\sigma^2}}e^{ik_0x}$$ My calculation is $$\int_{-\infty}^\infty ...
2
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1answer
87 views

Fourier series with a weighted mean square norm

I am interested in Fourier series with a non-uniformly weighted error norm. What I mean by this is that the usual Fourier series of a periodic function is a minimizer of the mean squared error: $$ J_N ...
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106 views

(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
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36 views

Basic question about the Discerete Fourier Transfrom

I have trouble understanding the transition from the infinite integral of the Fourier transform $$ \mathcal{F}f(v) = \int^\infty_{-\infty}e^{ivk}f(k)dk $$ to the discrete version $$ \mathbf{F}f_n = ...
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48 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
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60 views

Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
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1answer
44 views

Please help with this Discrete fourier transform question

Consider the ODE $\frac {d^2u}{dx^2} + 2\pi\frac {du}{dx} + \frac 54\pi^2u = g(x)$ where g is a periodic fuction with period 1 given by $g(x) = e^{\pi x}$ , $ 0 \le x \lt 1$. It is desired to find ...
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1answer
145 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
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60 views

What can be observed by evaluating a polynomial at roots of order greater than the polynomial itself?

I have been reading through an algorithms book on the use of FFT for large number multiplication. An example it used to emphasize a point was: Evaluate the following polynomial at all roots of unity ...
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39 views

Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
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50 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
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1answer
23 views

how to prove translation identity: $(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x)$?

Let $m, f\in S(\mathbb R^{n})$=The Schwartz space. My question: How to prove: $$(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x);$$ where ...
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1answer
25 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
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1answer
34 views

testing out the formula. I don't know how to intepret this!

$$\frac{d^{2}u}{dt^{2}}-4\frac{d^{3}u}{dt dx^{2}}+3\frac{d^{4}u}{dx^{4}}=0$$ $$u(x,0) = f(x)$$ $$\frac{du}{dt}(x,0) = g(x)$$ Relevant equations The attempt at a solution First I use fourier ...
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1answer
5k views

Derivative of function using discrete fourier transform (MATLAB)

I'm trying to find the derivative of a function $f(t)$ using the discrete fourier transform. My end goal is to do so numerically, but I suppose if I have the theory down then the rest should follow ...
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1answer
90 views

Recommend Fourier Analysis Workbook or online examples

I am studying a graduate level course in Fourier Analysis, however my Functional Analysis background is extremely weak, I have also never met Lebesgue Integration and it has been a while since I ...
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1answer
116 views

Fundamental Solution to Laplace equation using inverse Fourier

I'm trying to do the following integral (which is an inverse Fourier transform): $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{e^{i (x_1\xi_1 + x_2\xi_2) }} {4\pi^2 (\xi_1^2 ...
3
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1answer
195 views

What do imaginary components of frequencies represent in Fourier?

I'm a Biology Ph.D. student with some math literacy (multivariable calc, discrete math, linear algebra, quantum chemistry) under my belt, so apologies if my knowledge is substandard for the work I'm ...
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1answer
65 views

Solve PDE with the Fourier transform

I have a problem with solving PDEs with the fourier trasform method when the function not depends only on x and t but also on the y variable. In particular, when I have to solve this equation ...
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153 views

Reconstruction formula for a function of moderate decrease

I want to show that, for a function $f$ of moderate decrease and $\hat{f}(\xi)$ supported in $I=[-\frac{1}{2},\frac{1}{2}]$, $$f(x) = \sum^{\infty}_{n=-\infty} f(n)K(x-n)$$ where $K(y) ...
3
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1answer
261 views

Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases} $$ A problem in Stein's Fourier Analysis asks ...
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1k views

Fourier Transform of $sgn(t)$

I'm trying to find the Fourier transform of $sgn(t)$ where $$ sgn(t) = \begin{cases} 1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0. \end{cases} $$ By definition, $$ X(\omega) = ...
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1answer
61 views

How to find the Fourier transform of $\mathbf 1_{[0,2\pi]}(x)\sin(x)$?

How does one find the Fourier transform of $f(x):=\mathbf 1_{[0,2\pi]}(x)\sin(x)$? I have tried to use the definition from my text: \begin{align*} \hat f(\xi) & = \frac{1}{\sqrt{2\pi}} ...
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63 views

How to calculate this basic Fourier Transform?

I am trying to calculate the Fourier Transform of $g(t)=e^{-\alpha|t|}$, where $\alpha > 0$. Because there's an absolute value around $t$, that makes $g(t)$ an even function, correct? If that's ...
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2answers
13k views

fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
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1answer
46 views

Integral from inverse Fouriertransform of 1/(1+p^2)^2

In a calculation I end up with the following integral $$\int_0 ^\infty \frac{p \sin (pr)}{(1+p^2)^2}dp , $$ could someone give me a hint how to evaluate this one? (This integral comes from the ...
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1answer
66 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
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Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
2
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247 views

Inverse Fourier transform of absolute value of a given Fourier transform

Given that, for some $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, we have the Fourier transform of $f$ given as \begin{equation} F(k):=\int_{-\infty}^\infty ...
2
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55 views

Asymptotic behaviour of oscillating integral

I'm interested in the big $x$ ($x \to \infty$) behaviour for the following integral $\int_{-\infty}^{\infty} \frac{dk}{\sqrt{k^2+1}} \frac{e^{-\sqrt{k^2+1}/2}}{1-e^{-\sqrt{k^2+1}}} e^{ikx}$ After a ...
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1answer
63 views

Using the Duality of the Fourier Transform

How would I use the duality of the Fourier transform to find the inverse Fourier transform of $$ X(\omega) = 2\pi e^{a\omega} u(-\omega)? $$
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1answer
195 views

Fourier Series/Parseval's Theorem

I have pretty much completed this question and have found the Fourier representation to be; $$ f(x) =\frac A2 +\sum_{n=0}^\infty 2A\frac{\cos(((2n-1)(\pi x))/2f_o)}{\pi(2n-1)} $$ Now I don't ...
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1answer
24 views

Existence of a subsequence and convergence to $0$ of function solving heat equation

Let $f^j(x)$ be a sequence of integrable functions on the circle such that $$\int_{-\pi}^{\pi}|f^j (x) |^2 dx = 1.$$ ALso, let $u^j(x,t)$ solve the heat equation on the circle with initial data ...
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1k views

Why would you expand a square wave in a Fourier series?

The periodic square wave $$ f(x) = \cases{ 1 \text{ if } 0 \le x \le \pi \\ 0 \text { if } -\pi \le x < 0} $$ seems easy enough to work with. Why transform it into a series of sines and cosines? ...